Approximating spectral densities of large matrices

Abstract

In physics, it is sometimes desirable to compute the so-called \emph{Density Of States} (DOS), also known as the \emph{spectral density}, of a real symmetric matrix $A$. The spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues of $A$. But this approach is generally costly and wasteful, especially for matrices of large dimension. There exists alternative methods that allow us to estimate the spectral density function at much lower cost. The major computational cost of these methods is in multiplying $A$ with a number of vectors, which makes them appealing for large-scale problems where products of the matrix $A$ with arbitrary vectors are relatively inexpensive. This paper defines the problem of estimating the spectral density carefully, and discusses how to measure the accuracy of an approximate spectral density. It then surveys a few known methods for estimating the spectral density, and proposes some new variations of existing methods. All methods are discussed from a numerical linear algebra point of view.